Transportation Problem and Related Topics

2 Ardavan Asef-Vaziri June-2013Transportation Problem and Related Topics There are 3 plants, 3 warehouses. Production of Plants 1, 2, and 3 are 100, 150, 200 respectively. Demand of warehouses 1, 2 and 3 are 170, 180, and 100 units respectively. Transportation costs for each unit of product is given below Transportation problem : Narrative representation Warehouse Plant Formulate this problem as an LP to satisfy demand at minimum transportation costs.

3 Ardavan Asef-Vaziri June-2013Transportation Problem and Related Topics Plant 1 Warehouse 1 Plant 2 Plant 3 Warehouse 2Warehouse 3 Data for the Transportation Model Quantity demanded at each destination Quantity supplied from each origin Cost between origin and destination

4 Ardavan Asef-Vaziri June-2013Transportation Problem and Related Topics $12 $11 $13 $12 Plant 1Plant 2Plant 3 Warehouse 1Warehouse 2Warehouse 1 $14 $16 $12 $11 $15 Supply Locations Demand Locations Data for the Transportation Model

5 Ardavan Asef-Vaziri June-2013Transportation Problem and Related Topics Transportation problem I : decision variables x 11 x x 13 x 21 x 31 x 22 x 32 x 23 x 33

6 Ardavan Asef-Vaziri June-2013Transportation Problem and Related Topics Transportation problem I : decision variables x 11 = Volume of product sent from P1 to W1 x 12 = Volume of product sent from P1 to W2 x 13 = Volume of product sent from P1 to W3 x 21 = Volume of product sent from P2 to W1 x 22 = Volume of product sent from P2 to W2 x 23 = Volume of product sent from P2 to W3 x 31 = Volume of product sent from P3 to W1 x 32 = Volume of product sent from P3 to W2 x 33 = Volume of product sent from P3 to W3 Minimize Z = 12 x x x x x x x x x 33

7 Ardavan Asef-Vaziri June-2013Transportation Problem and Related Topics Transportation problem I : supply and demand constraints: equal only of Total S = Total D x 11 + x 12 + x 13 = 100 x 21 + x 22 + x 23 =150 x 31 + x 32 + x 33 = 200 x 11 + x 21 + x 31 = 170 x 12 + x 22 + x 32 = 180 x 13 + x 23 + x 33 = 100 x 11, x 12, x 13, x 21, x 22, x 23, x 31, x 32, x 33  0

8 Ardavan Asef-Vaziri June-2013Transportation Problem and Related Topics Transportation problem I : supply and demand constraints: ≤ for S, ≥ for D always correct x 11 + x 12 + x 13 ≤ 100 x 21 + x 22 + x 23 ≤ 150 x 31 + x 32 + x 33 ≤ 200 x 11 + x 21 + x 31 ≥ 170 x 12 + x 22 + x 32 ≥ 180 x 13 + x 23 + x 33 ≥ 100 x 11, x 12, x 13, x 21, x 22, x 23, x 31, x 32, x 33  0

9 Ardavan Asef-Vaziri June-2013Transportation Problem and Related Topics Origins We have a set of ORIGINs Origin Definition: A source of material - A set of Manufacturing Plants - A set of Suppliers - A set of Warehouses - A set of Distribution Centers (DC) In general we refer to them as Origins m 1 2 i s1s2sisms1s2sism There are m origins i=1,2, ………., m Each origin i has a supply of s i

10 Ardavan Asef-Vaziri June-2013Transportation Problem and Related Topics Destinations We have a set of DESTINATIONs Destination Definition: A location with a demand for material - A set of Markets - A set of Retailers - A set of Warehouses - A set of Manufacturing plants In general we refer to them as Destinations n 1 2 j d1d2didnd1d2didn There are n destinations j=1,2, ………., n Each origin j has a supply of d j

11 Ardavan Asef-Vaziri June-2013Transportation Problem and Related Topics There is only one route between each pair of origin and destination Items to be shipped are all the same for each and all units sent from origin i to destination j there is a s hipping cost of C ij per unit Transportation Model Assumptions

12 Ardavan Asef-Vaziri June-2013Transportation Problem and Related Topics C ij : cost of sending one unit of product from origin i to destination j m 1 2 i n 1 2 j C 1n C 12 C 11 C 2n C 22 C 21 Use Big M (a large number) to eliminate unacceptable routes and allocations.

13 Ardavan Asef-Vaziri June-2013Transportation Problem and Related Topics X ij : Units of product sent from origin i to destination j m 1 2 i n 1 2 j x 1n x 12 x 11 x 2n x 22 x 21

14 Ardavan Asef-Vaziri June-2013Transportation Problem and Related Topics The Problem m 1 2 i n 1 2 j The problem is to determine how much material is sent from each origin to each destination, such that all demand is satisfied at the minimum transportation cost

15 Ardavan Asef-Vaziri June-2013Transportation Problem and Related Topics The Objective Function m 1 2 i n 1 2 j If we send X ij units from origin i to destination j, its cost is C ij X ij We want to minimize

16 Ardavan Asef-Vaziri June-2013Transportation Problem and Related Topics Transportation problem I : decision variables x 11 x x 13 x 21 x 31 x 22 x 32 x 23 x 33

17 Ardavan Asef-Vaziri June-2013Transportation Problem and Related Topics Transportation problem I : supply and demand constraints x 11 + x 12 + x 13 =100 +x 21 + x 22 + x 23 =150 +x 31 + x 32 + x 33 =200 x 11 + x 21 + x 31 =170 x 12 + x 22 + x 32 =180 x 13 + x 23 + x 33 = 100 In transportation problem. each variable Xij appears only in two constraints, constraints i and constraint m+j, where m is the number of supply nodes. The coefficients of all the variables in the constraints are 1.

18 Ardavan Asef-Vaziri June-2013Transportation Problem and Related Topics Our Task Our main task is to formulate the problem. By problem formulation we mean to prepare a tabular representation for this problem. Then we can simply pass our formulation ( tabular representation) to EXCEL. EXCEL will return the optimal solution. What do we mean by formulation?

19 Ardavan Asef-Vaziri June-2013Transportation Problem and Related Topics Cost Table

20 Ardavan Asef-Vaziri June-2013Transportation Problem and Related Topics Right Hand Side (RHS)

21 Ardavan Asef-Vaziri June-2013Transportation Problem and Related Topics Left Hand Side (RHS), and Objective Function

22 Ardavan Asef-Vaziri June-2013Transportation Problem and Related Topics ≤ for Supply, ≥ for Demand unless Some Equality Requirement is Enforced

23 Ardavan Asef-Vaziri June-2013Transportation Problem and Related Topics ≤ for Supply, ≥ for Demand unless Some Equality Requirement is Enforced

24 Ardavan Asef-Vaziri June-2013Transportation Problem and Related Topics Optimal Solution Extra Credit. How the colors were generated and what they mea? Using Conditional formatting. Green if the decision variable is >0 Red if the constraint is binding LHS = RHS

25 Ardavan Asef-Vaziri June-2013Transportation Problem and Related Topics Example: Narrative Representation We have 3 factories and 4 warehouses. Production of factories are 100, 200, 150 respectively. Demand of warehouses are 80, 90, 120, 160 respectively. Transportation cost for each unit of material from each origin to each destination is given below. Destination Origin Formulate this problem as a transportation problem

26 Ardavan Asef-Vaziri June-2013Transportation Problem and Related Topics Excel : Data

27 Ardavan Asef-Vaziri June-2013Transportation Problem and Related Topics 11 repairmen and 10 tasks. The time (in minutes) to complete each job by each repairman is given below. Assign each task to one repairman in order to minimize to total repair time by all the repairmen. In the assignment problem, all RHSs are 1. That is the only difference with the transportation problem,. The Assignment Problem : Example

28 Ardavan Asef-Vaziri June-2013Transportation Problem and Related Topics The Assignment Problem : Example